Woodin cardinals and the Continuum Hypothesis

[Jim__]

The Zermelo Fraenkel axioms lead to many possible models of set theory. In some of these models, the Continuum Hypothesis is true, in others, it's false. A mathematician, Hugh Wooden, has proposed a method for resolving this issue. A method that allows moving to a higher plane and on this higher plane, only some of these models would be valid; and in all those models, the Continuum Hypothesis is true. This higher plane is based on Gödel's "constructibale universe".

The August 1 issue of New Scientist has a pretty good article on this. An excerpt:

...

For most purposes, life within these structures is the same: most everyday mathematics does not differ between them, and nor do the laws of physics. But the existence of this mathematical "multiverse" also seemed to dash any notion of ever getting to grips with the continuum hypothesis. As Cohen was able to show, in some logically possible worlds the hypothesis is true and there is no intermediate level of infinity between the countable and the continuum; in others, there is one; in still others, there are infinitely many. With mathematical logic as we know it, there is simply no way of finding out which sort of world we occupy.

That's where Hugh Woodin of the University of California, Berkeley, has a suggestion. The answer, he says, can be found by stepping outside our conventional mathematical world and moving on to a higher plane.

Woodin is no "turn on, tune in" guru. A highly respected set theorist, he has already achieved his subject's ultimate accolade: a level on the infinite staircase named after him. This level, which lies far higher than anything envisaged in Gödel's L, is inhabited by gigantic entities known as Woodin cardinals.

Woodin cardinals illustrate how adding penthouse suites to the structure of mathematics can solve problems on less rarefied levels below. In 1988 the American mathematicians Donald Martin and John Steel showed that if Woodin cardinals exist, then all "projective" subsets of the real numbers have a measurable size. Almost all ordinary geometrical objects can be described in terms of this particular type of set, so this was just the buttress needed to keep uncomfortable apparitions such as Banach and Tarski's ball out of mainstream mathematics.

much more ...

[WheelWalker]

1. In the immortal words of a pointy-eared science officer...

http://www.youtube.com/watch?v=cFods1KSWsQ

[bananas]

2. I have embraced the diversity of the mathematical multiverse

Others are less convinced. Hamkins, who is a former student of Woodin's, holds to the idea that there simply are as many legitimate logical constructions for mathematics as we have found so far. He thinks mathematicians should learn to embrace the diversity of the mathematical multiverse, with spaces where the continuum hypothesis is true and others where it is false. The choice of which space to work in would then be a matter of personal taste and convenience. "The answer consists of our detailed understanding of how the continuum hypothesis both holds and fails throughout the multiverse," he says.

[Jim__]

5. My understanding of what the article says about Ultimate L is that ....

the mathematical universes where the continuum hypothesis doesn't hold, are actually contained within a larger universe where it does hold:

Woodin and others spotted the germ of a new, more radical approach while investigating particular patterns of real numbers that pop up in various L-type worlds. The patterns, known as universally Baire sets, subtly changed the geometry possible in each of the worlds and seemed to act as a kind of identifying code for it. And the more Woodin looked, the more it became clear that relationships existed between the patterns in seemingly disparate worlds. By patching the patterns together, the boundaries that had seemed to exist between the worlds began to dissolve, and a map of a single mathematical superuniverse was slowly revealed. In tribute to Gödel's original invention, Woodin dubbed this gigantic logical structure "ultimate L".

I don't believe the article contains enough information to actually decide. But my understanding is, all the mathematical diversity holds; but it's all contained within a larger universe.

[agent46]

3. So cool...

but it doesn't mean what you think it means.

[struggle4progress]

4. The large cardinal hypotheses are interesting to logicians, but really! Godel's "V=L"

is much more natural for the working mathematician and implies AC + CH